Method and apparatus for providing an energy-based signal tracking loop

ABSTRACT

A method and apparatus for tracking a signal comprises correlating a digital signal with a code using a hypothesis at a plurality of frequencies and at least one delay to produce correlation results, measuring an energy value of the correlation results, adjusting at least one of the frequency or delay in response to the measured energy value to form the hypothesis.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation-in-part of co-pending U.S. patentapplication Ser. No. 11/716,118, filed Mar. 9, 2007, which is acontinuation-in-part of co-pending U.S. patent application Ser. No.10/690,973 filed Oct. 22, 2003, which is herein incorporated byreference.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The following generally relates to a digital receiver and, moreparticularly, to a method and apparatus for providing an energy basedsignal tracking loop within a digital receiver.

2. Description of the Related Art

A global navigation satellite system (GNSS) typically includes multipleorbiting satellites and at least one receiver (GNSS receiver). Examplesof GNSS include the global positioning system (GPS), the Galileopositioning system (GALILEO), the Global'naya NavigatsionnayaSputnikovaya Sistema (GLONASS), and the like.

In operation, each of the satellites broadcast a radio signal (GNSSsignal) upon which is modulated certain information. The GNSS receivercaptures the broadcast satellite signal, extracts the informationmodulated upon the signal, and computes an estimate of the position ofthe GNSS receiver using the information. More specifically, the receiverposition is determined by computing, for each satellite in view of theGNSS receiver, a time delay between the time of the transmission fromthe satellite and a time of reception of the satellite signal at theGNSS receiver. The time delay multiplied by the speed of light providesa distance (a pseudorange) from the GNSS receiver to the satellite.Using pseudoranges to a number of satellites, the receiver computes theposition.

To enable the receiver to extract information from the GNSS signals, thecarrier frequency and the code delay (which is indicative of thepseudorange) must be determined and tracked. The GNSS signal ismodulated with a pseudorandom (PN) code that is correlated with a PNcode in the receiver. The correlation process produces an estimate offrequency and code delay.

These estimates vary over time due to motion of the receiver relative tothe satellites, satellite motion through space, clock inaccuracies, andthe like. As such, time and frequency tracking loops are used. Theseloops utilize a discriminator that derives a time and/or frequency errorand applies the error to the frequency or timing loop to compensate forthe error. Such loops (e.g., phase or frequency lock loops) are wellknown and are widely used in GNSS receivers.

One issue that arises with the use of conventional phase or frequencytracking loops is that they rely upon a discriminator, that uses anonlinearity. Such discriminators are not sensitive to low-signalstrength signals. Consequently, at low-signal strength, conventionalsignal tracking loops fail.

Therefore, there is a need in the art for a tracking loop that tracksdelay and frequency of low-signal strength signals.

SUMMARY OF THE INVENTION

Embodiments of the invention comprise a method and apparatus fortracking a signal comprising correlating a digital signal with a codeusing a hypothesis at a plurality of frequencies and at least one delayto produce correlation results, measuring energy value of thecorrelation results, adjusting at least one of frequency or delay inresponse to the measured energy value to form the hypothesis.

BRIEF DESCRIPTION OF THE DRAWINGS

So the manner which the above recited features are attained and can beunderstood in detail, a more detailed description is described belowwith reference to figures illustrated in the appended drawings.

The figures in the appended drawings, like the detailed description, areexamples. As such, the figures and the detailed description are not tobe considered limiting, and other equally effective examples arepossible and likely. Furthermore, like reference numerals in the figuresindicate like elements, and wherein:

FIG. 1 is a block diagram of a GNSS receiver as used in a location basedservices platform;

FIG. 2 depicts a block diagram of a GNSS receiver utilizing acoprocessor for processing correlation information;

FIG. 3 depicts a block diagram of one embodiment of the presentinvention;

FIG. 4 depicts a graph of frequency versus delay in accordance with thepresent invention;

FIG. 5 depicts a flow diagram of an operation of a discriminator forproducing a measured energy vector in accordance with one embodiment ofthe present invention; and

FIG. 6 depicts a flow diagram of a method of operation of one embodimentof a filter used for generating correction vectors for frequency anddelay correction in accordance with one embodiment of the presentinvention.

DETAILED DESCRIPTION

FIG. 1 depicts a block diagram of a mobile device 100 incorporating anembodiment of the present invention. The mobile device 100 comprises aGNSS receiver 102 and a location based services (LBS) platform 104. Theuse of a GNSS receiver within which the invention is incorporated formsone application of the invention. Other platforms that require signalcorrelation may find use of the present invention.

The LBS platform 104 may comprise any computing device that is capableof executing location based services (LBS) application software such as,but not limited to, cellular telephone circuitry, a personal digitalassistant device, a pager, a laptop computer, a computer in anautomobile, and the like. The LBS platform 104 comprises a centralprocessing unit (CPU) 118, support circuits 122, memory 120, and LBSplatform circuits 124. The CPU 118 may comprise one or more well knownmicroprocessors or microcontrollers. The support circuits 122 are wellknown circuits that support the operation and functionality of the CPU118. The support circuits 122 may comprise clock circuits, input/outputcircuits, power supplies, cache, and the like. The memory 120 maycomprise random access memory, read only memory, disc drives, removablememory, combinations thereof, and the like. The memory generally storesan operating system 126 and software 128 such as LBS applications that,when executed by the CPU, utilize position information supplied by theGNSS receiver 102 to provide various services to a user.

The LBS platform 104 may further include LBS platform circuits 124 thatprovide specific functions to the LBS platform. For example, the LBSplatform circuits 124 may include a cellular telephone transceiver, anetwork interface card for coupling data to or from a computer network,a display, and/or other circuits that provide LBS platformfunctionality.

The GNSS receiver 102 receives satellite signals, correlates the signalswith a locally generated code, and uses the correlation results todetermine the position of the receiver. More specifically, the receiver102 comprises an antenna 106, a receiver front-end 132, which includes aradiofrequency-to-intermediate frequency (RF/IF) converter 108, and ananalog-to-digital (A/D) converter 110, a correlation circuit 112, acorrelation processor 114, and interface logic 116.

Signals (such as GNSS signals) are received by the antenna 106. TheRF/IF converter 108 filters, amplifies, and frequency shifts the signalsfor digitization by the (A/D) converter 110. The elements 106, 108, and110 are substantially similar to those elements used in a conventionalGNSS or assisted GNSS receiver. These elements generally form what isknown as the front-end 132 of a receiver.

The output of the A/D 110 is coupled to a correlation circuit 112. Inone embodiment, the correlation circuit 112 comprises a multichannelcorrelator (e.g., n correlation channels represented by correlators 130₁, 130 ₂, . . . 130 _(n) where n is an integer) that creates a series ofcorrelation results. The correlators 130 ₁, 130 ₂, . . . 130 _(n) areherein collectively referred to as correlators 130. One suchillustrative correlation circuit is described in commonly assigned U.S.Pat. Nos. 6,606,346 and 6,819,707, which are incorporated herein byreference in their entireties. The correlation circuits described inthese patents represent one of many types of correlation circuits thatcould be used to correlate the received satellite signal (or portionthereof) with a locally generated code. Any correlation circuit thatproduces a series of correlation results can be used as a component ofthe present invention.

The correlation results (a stream that is generated at a first rate) areprocessed, in real time, by the correlation processor 114. Thecorrelation processor 114 stores and processes sets of correlationresults to rapidly estimate received signal parameters that may be usedto tune the correlation circuit 112 to acquire the satellite signals,e.g., the correlation processor 114 performs a two-dimensional searchregarding Doppler frequency and/or bit timing. The correlation processor114 accesses and processes the correlation results at a second rate inone embodiment of the invention, the second rate is faster than thefirst rate such that the stream of correlation results from all thecorrelators 130 can be repeatedly processed in real time withoutimpacting the operation of the correlation circuit 112. Thisinformation, single correlation processor 114 may process multiplechannels of correlation results.

The interface logic 116 couples data and control signals between the LBSplatform 104 and the GNSS receiver 102. The CPU 118 generates controlsignals that request the GNSS receiver 102 to start up and acquire thesatellite signals. The processed signals may be coupled to the CPU 118for further processing or transmission to a remote location (a locationserver) for further processing. The use of the acquired satellitesignals to determine the position of the receiver is disclosed in U.S.Pat. No. 6,453,237, which is herein incorporated by reference in itsentirety.

FIG. 2 depicts a block diagram of the correlation processor 114. Thecorrelation processor 114 comprises a correlation history buffer 202, acoprocessor 204, a microcontroller 210, and support circuits 208. Thecorrelation results (a data stream) are coupled to the correlationhistory buffer 202, a memory having a length, for example, of 1000samples per channel (e.g., 1 second of GPS data, where the samples arecreated in 1 millisecond intervals). The buffer may be any form ofrepeated use memories such as a circular buffer, a ping-pong buffer, andthe like. The buffer stores a history of correlation results as theresults are generated at a first rate by one or more correlators 130 ofFIG. 1.

The history is accessed by the coprocessor 204 and processed at leastonce, and in all likelihood, many times to assist in rapidly tuning thecorrelation circuit. A single coprocessor 204 may process correlationresults for a number of channels. The coprocessor 204 is supported bysupport circuits 208 comprising, for example, cache, power supplies,clock circuits, and the like. The coprocessor is also coupled to amemory 208 comprising random access memory, read only memory, and/or acombination thereof. The memory comprises correlation processingsoftware 212 (instructions) that, when executed by the coprocessor 204,enhance the acquisition of satellite signals by the receiver.

In response to a request from the CPU 118 for specific information fromthe coprocessor, the microcontroller 210 provides program selection andsequencing signals to the coprocessor 204. Specifically, themicrocontroller 210 selects the instruction set that is to be executedby the coprocessor 204 to fulfill the request from the CPU 118. Themicrocontroller 210 provides sequencing signals to the coprocessor 204to step the coprocessor through the instruction set.

The coprocessor 204 and microcontroller 210 are coupled to the CPU 118of the LBS platform 100 via the interface logic 116. As such, the CPU118 requests information from the GNSS receiver 102 via the interfacelogic 116. The results of the coprocessor computation are coupled to theCPU 118 through the interface logic 116. In this manner, the CPU 118 canrequest information and then go on to perform other processes while thecoprocessor performs the signal processing function. Thus, the CPU 118is not burdened by the computation of GNSS signals.

FIG. 3 depicts a functional block diagram depicting the operation of atracking loop 300 comprising the correlators 130, the correlationhistory buffer 202, a discriminator 310, and a feedback filter 312. Inthis embodiment, the correlators 130 produce a series of correlationresults that are buffered within the correlation history buffer 202. Thediscriminator 310 accesses the buffered correlation information;correlation data processes that information to form a measured energyvector. The measured energy vector is applied to a feedback filter 312that produces frequency and delay estimates that are coupled to thecorrelators 130. The term energy (or alternatively, magnitude oramplitude) are produced by any non-linear operation that creates aquantity that is sensitive to received signal power.

In the depicted embodiment, there are five correlators operating inparallel on a given input signal. These correlators are early in time,late in time, high frequency, low frequency, and punctual. Each of thecorrelators produces a specific correlation output that is controlled bya frequency and delay estimate. For example, the early time correlator300 and the late time correlator 302 operate at the same frequencyestimate but have a delay estimate that is either early in time or latein time compared to the center expected time or punctual delay of thecorrelator 308. Similarly, the high frequency correlator 304, the lowfrequency correlator 306 operate at the same delay but operate withdifferent frequency estimates for each correlator. The high and lowfrequencies are respectfully higher or lower than an expected frequencyor the frequency used by the punctual correlator 308. Although eachcorrelator operates on a single input signal, the outputs will varydepending on the frequency and delay estimates that are applied to eachof the correlators. The correlation outputs are temporarily stored inthe correlation history buffer that is then accessed by thediscriminator 310.

The discriminator 310 is implemented by a software routine executingupon the coprocessor 204 of FIG. 2. In other embodiments, thediscriminator 310 operates in real time on the outputs from thecorrelators 130 without utilizing a correlation history buffer. Thediscriminator 310 produces an energy level representing the energy ineach correlator output. In one embodiment of the invention, the latecorrelation value is subtracted from the early correlation value tocreate a E-L correlation value. Similarly, the low frequency correlationvalue is subtracted form the high frequency correlation value to createan H-L correlation value. This reduces the computation space andsimplifies the energy computation. The E-L correlation value (ameasurement indicative of delay offset), the H-L correlation value (ameasurement indicative of frequency offset) and a magnitude value areapplied the feedback filter 312 as components of a vector. This measuredvector is represented by these three measured quantities. These measuredquantities within the measured energy vector are applied to the feedbackfilter 312, which updates a correction vector that is applied to thecorrelators 130. The correction vector adjusts the frequency and delayestimates used to create the correlation samples in an attempt tooptimize the correlation process, e.g., improve the magnitude of themeasured vector at a center frequency and delay value.

FIG. 4 graphically depicts the process used by the tracking loop 300 ofFIG. 3. The multidimensional graph 400 having a first dimension alongthe frequency axis 402, a second dimension along the delay axis 404 andan amplitude in the Z axis 420 (extending out of the plane of the page).An early sampling location 410, a late sampling location 412, ahigh-frequency sampling location 406, and a low-frequency samplinglocation 408 are depicted on the graph 400. A typical multidimensionalsample that is represented by an energy vector 418 occurs at correlationoutput 416. The punctual sample is at 414. In operation, the measuredenergy vector 418 represents the correlation output 416 as amultidimensional measurement. The current estimates of frequency,amplitude and power updated by the tracking loop 300 to have a maximumamplitude value extending from the plane of the page and located at thepunctual sampling location 414. The measured energy vector is used toadjust the delay and frequency of the correlation process performedwithin the correlators 130 to improve the energy vector.

FIG. 5 depicts a flow diagram of a method 500 of operation of thediscriminator 310. The method 500 begins at step 502 and proceeds tostep 504, where the correlators correlate input signals at a pluralityof delay and frequency hypotheses. These correlation values may beanalysed in real or stored as a correlation history as depicted. At step506, the discriminator generates a measured energy vector containingpresent states. The routine ends at step 508. This method 500 isrepeatedly utilized to generate a measured energy vector for each timeperiod of the use by the correlation history buffer.

In generating the correlation data that is used by the discriminator,the depicted embodiment in FIG. 3 utilizes five correlators in parallel.Alternatively, a fewer number of correlators could be used and the datacollected in the correlation history buffer may be manipulated by thecoprocessor in performing the discriminator function to createcorrelation data at various delays and frequency hypotheses withoutactually controlling the correlator for each of those data points. Inparticular, values of correlation for offset frequencies (e.g., highfrequency correlator and low frequency correlator) can be generated froma correlation history generator by a correlator operating at a singlefrequency. The utilization of a coprocessor in this manner is describedin commonly assigned U.S. patent application Ser. No. 10/690,973 filedOct. 22, 2003 which is incorporated by reference herein in its entirety.

FIG. 6 depicts a flow diagram of the operation of a feedback filter inaccordance with one embodiment of the present invention. The method 600begins at step 602 and proceeds to step 604, where the filter isinitialized. At step 606, the method 600 applies the measured energyvector to the filter. At step 608, the method 600 determines frequencyand delay utilizing the measured energy vector. At step 610, the method600 generates a frequency and delay correction vector that is then sentto the correlators to be used to produce a new set of correlationvalues. At step 612, the method 600 queries whether an another vector isto be computed. If step 612 is affirmatively answered, then the methodproceeds to step 606 and applies the new measured energy vector to thefilter. If the query at step 612 is negatively answered, the method endsat step 618.

In the manner described above, the measured energy vector is applied toa fixed gain filter to produce an error value that is then used forgenerating a new frequency and delay correction vector. Although such aprocess facilitates the use of a measured energy vector in an energybased tracking loop to adjust the delay and frequency parameters of thecorrelators and optimize the sampling or the correlations, alternativelya dynamic model can be utilized. Since the variables (delay, frequency,amplitude) form a multi-dimensional problem where the variables areinterdependent, the use of a dynamic model is well suited to estimatingthe delay and frequency to be used by the correlators to improve theenergy vector.

To use a dynamic model, at step 614, the measured energy vector isapplied to the filter that utilizes a plurality of states within adynamic model. Such a dynamic model may be created using an extendedKalman filter. Extended Kalman filters are generally well known in theart and the use of them in providing a dynamic basis for filtering iswell known. In this application, the energy vector provides the measuredquantities that are used to alter the states of the dynamic model. Thesestates include a frequency value, a delay value and, optionally, anamplitude value (equivalently, an energy or magnitude value). Theamplitude state may be fixed, variable or provided from outside of thedynamic mode. In that sense, the amplitude state is an optional state ofthe dynamic model. Once the states within the dynamic model are updatedutilizing the latest measured energy vector, the method 600 proceeds tostep 616 wherein the dynamic model projects the states into the futureto create a correction vector that will produce optimal states in thefuture. The correction vector is then applied to the correlators toproduce a new measured energy vector.

By using the energy vector as a basis for performing frequency and delayadjustments for the correlators, the embodiments of the inventionoperate at extremely low input signal levels, e.g., a GNSS satellitesignal can be tracked at signal strengths of up to −160 dBm. Whetherusing a static or dynamic model, the adjustments to delay and frequencyare intended to maximize the energy (or magnitude) value. With eachiteration of the method, the method attempts to improve the energyvalue.

The following is a description of one embodiment of a dynamic modelbased upon an extended Kalman filter that can be used to facilitate anenergy based tracking loop as described above.

I. Model

A. State Vector

The state vector for the kth block of correlation values is specifiedas:x_(k)=[τ_(k),ν_(k),a_(k)]^(t) , a _(k)=10 log₁₀(A _(k))  (1)where [•]′ denotes vector transpose, ν_(κ) denotes velocity (frequency)in [m/sec], r_(κ) denotes a pseudorange in [m], and ακ (Aκ) denotes thesignal amplitude in [dB] (Volts) all at the end of the κth block. Theblock interval is T sec.

The state vector is updated according to a constant velocity, constantsignal amplitude, random walk type model: $\begin{matrix}{{x_{k + 1} = {{\Phi\quad x_{k}} + w_{k}}},{\Phi = \begin{bmatrix}1 & T & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}}} & (2)\end{matrix}$with the process noise is modeled as: $\begin{matrix}{{w_{k} = {\begin{bmatrix}{{\overset{\_}{\alpha}}_{k}\frac{T^{2}}{2}} \\{{\overset{\_}{\alpha}}_{k}T} \\{\overset{\_}{a}}_{k}\end{bmatrix} = {\overset{IID}{\sim}{\mathcal{N}( {0,Q} )}}}},{Q = \begin{bmatrix}{\sigma_{\overset{\_}{\alpha}}^{2}\frac{T^{4}}{4}} & {\sigma_{\overset{\_}{\alpha}}^{2}\frac{T^{3}}{2}} & 0 \\{\sigma_{\overset{\_}{\alpha}}^{2}\frac{T^{3}}{2}} & {\sigma_{\overset{\_}{\alpha}}^{2}T^{2}} & 0 \\0 & 0 & \sigma_{\overset{\_}{a}}^{2}\end{bmatrix}}} & (3)\end{matrix}$where $\sigma\frac{2}{\alpha}\quad{and}\quad\sigma\frac{2}{\alpha}$are variances associated with white, zero mean, discrete time randomacceleration and random signal-amplitude-change terms α_(κ) and a_(κ) ,respectively. Also, N(μ,K) denotes the normal distribution of specifiedmean and covariance. Lastly, 0 denotes a column vector of zeros ofappropriate dimension.

An alternative, constant-acceleration model may also be considered.Here, the state vector is augmented to include an acceleration term:$\begin{matrix}{{x_{k} = \lbrack {T_{k},v_{k},\alpha_{k},a_{k}} \rbrack^{\prime}},{\Phi = \begin{bmatrix}1 & T & {\frac{1}{2}T^{2}} & 0 \\0 & 1 & T & 0 \\0 & 0 & 1 & 0 \\0 & 0 & 0 & 1\end{bmatrix}}} & (4)\end{matrix}$with the process noise is modeled as: $\begin{matrix}{{w_{k} = {\begin{bmatrix}{{\overset{\_}{\alpha}}_{k}\frac{T^{2}}{2}} \\{{\overset{\_}{\alpha}}_{k}T} \\{\overset{\_}{\alpha}}_{k} \\{\overset{\_}{a}}_{k}\end{bmatrix} = {\overset{IID}{\sim}{\mathcal{N}( {0,Q} )}}}},{Q = \begin{bmatrix}{\sigma_{\overset{\_}{\alpha}}^{2}\frac{T^{4}}{4}} & {\sigma_{\overset{\_}{\alpha}}^{2}\frac{T^{a}}{2}} & {\sigma_{\overset{\_}{\alpha}}^{2}\frac{T^{a}}{2}} & 0 \\{\sigma_{\overset{\_}{\alpha}}^{2}\frac{T^{3}}{2}} & {\sigma_{\overset{\_}{\alpha}}^{2}T^{2}} & {\sigma_{\overset{\_}{\alpha}}^{2}T} & 0 \\{\sigma_{\overset{\_}{\alpha}}^{2}\frac{T^{2}}{2}} & {\sigma_{\overset{\_}{\alpha}}^{2}T} & \sigma_{\overset{\_}{\alpha}}^{2} & 0 \\0 & 0 & 0 & \sigma_{\overset{\_}{a}}^{2}\end{bmatrix}}} & (5)\end{matrix}$where α_(κ) is an unmodeled acceleration term modeled as a realizationof a zero mean, white random process of variance$\sigma{\frac{2}{\alpha}.}$B. Measurement Vector

The measurement vector for the kth block is defined as:z_(k)=[z_(k) ^(EL),z_(k) ^(HL),z_(k) ^(P)]^(t)   (6)where z_(k) ^(EL) is an early-minus-late delay type measurement, z_(k)^(HL) is a high-minus-low frequency type measurement, and z_(k) ^(P) isa prompt signal amplitude type measurement. In the current context, themeasurements are a nonlinear function of the state vector:z _(k) =h(x _(k))+n _(k)  (7)where ν_(κ) is the measurement noise, and h: R^(3×1)→R^(3×1) is anonlinear mapping with R^(3×1) is the set of real, three element columnvectors.

Two possible forms of this nonlinearity are now described.

B.1 EGY COP Based Measurements

The mean of the early-delay-minus-late-delay measurement isapproximately linear in pseudorange over the interval r_(k) ^(a)±Δ_(r)where Δ_(r) is the spacing between correlation vector delays, and r_(k)^(a) is the prompt acquisition range for the kth block. Likewise, themean of the high-frequency-minus-low-frequency measurements isapproximately linear in frequency over an interval ν_(k) ^(a)±Δ_(ν),Δ_(ν)<δ where δ is the high/low frequency (or velocity) offset, andν_(k) ^(a) is the acquisition velocity associated with the kth block.The dependence of the mean of these two measurements on signal amplitudeis well approximated by second order polynomials as is the promptmeasurement. Thus: $\begin{matrix}{{h( x_{k} )} = {{\begin{bmatrix}{h_{0}( x_{k} )} \\{h_{1}( x_{k} )} \\{h_{2}( x_{k} )}\end{bmatrix} \approx {\overset{\_}{h}( x_{k} )}} = {\lbrack \quad\begin{matrix}{( {r_{k} - r_{k}^{a}} ){\sum\limits_{i = 0}^{N_{0}}\quad{\eta_{i}a_{k}^{i}}}} \\{\quad{( {v_{k} - v_{k}^{a}} ){\sum\limits_{i = 0}^{N_{1}}\quad{\zeta_{i}a_{k}^{i}}}}} \\{\sum\limits_{i = 0}^{N_{2}}\quad{\psi_{i}a_{k}^{i}}}\end{matrix} \rbrack \approx {{h( {\quad\overset{\quad\_}{x}}_{\quad k} )} + {H_{\quad k}( {x_{\quad k} - {\quad\overset{\quad\_}{x}}_{\quad k}} )}}}}} & (8) \\{\quad{H_{k} = {{\frac{\partial h}{\partial x^{\prime}}{❘_{\quad}}_{x = {\overset{\_}{x}}_{k}}} = \begin{bmatrix}{\sum\limits_{i = 0}^{N_{0}}\quad{\eta_{i}a_{k}^{i}}} & 0 & {\quad{( {r_{k} - r_{k}^{a}} ){\sum\limits_{i = 0}^{N_{0}}\quad{i\quad\eta_{i}a_{k}^{i - 1}}}}} \\0 & {\sum\limits_{i = 0}^{N_{1}}\quad{\zeta_{i}a_{k}^{i}}} & {\quad{( {v_{k} - v_{k}^{a}} ){\sum\limits_{i = 0}^{N_{1}}\quad{i\quad\zeta_{i}a_{k}^{i - 1}}}}} \\0 & 0 & {\sum\limits_{i = 0}^{N_{2}}\quad{i\quad\psi_{i}a_{k}^{i - 1}}}\end{bmatrix}}}} & (9)\end{matrix}$where x_(k) and {tilde over (x)}_(k) are “close” in a Euclidean sense.Strictly speaking, acquisition frequency errors affect the pseudorangemeasurement, acquisition range errors affect the frequency measurement,and frequency and range errors both affect the signal amplitudemeasurement. The polynomial coefficients are designed to take smallerrors into account.

The measurement noise is modeled as a realization of a discrete time,zero mean, temporally uncorrelated, normal random process withcovariance generally dependent on the state vector. Monte-Carlosimulations have led to the following observations.

-   -   The individual measurement noise components are almost mutually        uncorrelated implying a diagonal measurement noise covariance.    -   The individual measurement noise variances are slowly changing,        approximately linear functions of signal amplitude in dB while        being essentially independent of pseudorange and velocity.        Mathematically, $\begin{matrix}        {\quad{{ n_{k} \sim{\mathcal{N}( {0,R_{k}} )}},{R_{k} = \begin{bmatrix}        \sigma_{{EL}_{k}}^{2} & 0 & 0 \\        0 & \sigma_{{HL}_{k}}^{2} & 0 \\        0 & 0 & \sigma_{P_{k}}^{2}        \end{bmatrix}}}} & (10)        \end{matrix}$        where: $\begin{matrix}        \begin{matrix}        {{\sigma_{{EL}_{k}}^{2} \approx {\overset{\_}{\sigma}}_{{EL}_{k}}^{2}} = {\sum\limits_{i = 0}^{K_{0}}\quad{\lambda_{i}a_{k}^{i}}}} \\        {{\sigma_{{HL}_{k}}^{2} \approx {\overset{\_}{\sigma}}_{{HL}_{k}}^{2}} = {\sum\limits_{i = 0}^{K_{1}}\quad{\beta_{i}a_{k}^{i}}}} \\        {{\sigma_{P_{k}}^{2} \approx {\overset{\_}{\sigma}}_{P_{k}}^{2}} = {\sum\limits_{i = 0}^{K_{2}}\quad{\gamma_{i}a_{k}^{i}}}}        \end{matrix} & (11)        \end{matrix}$        B.2 HSS COP Based Measurements        B.2.a Prompt. It will prove convenient to write the Early-Late        delay detector in matrix notation: $\begin{matrix}        \begin{matrix}        {z^{P} = {\sum\limits_{\ell = 0}^{L - 1}\quad{u_{\ell}^{H}B\quad u_{\ell}}}} \\        {{u_{\ell}\overset{IID}{\sim}{\mathcal{N}( {\mu_{u},K_{u}} )}},{\mu_{u} = {{{AG}( {r - r^{a}} )}{e( {v - v^{a}} )}}},{K_{u} = {\sigma^{2}I}}} \\        {{{G( {r - r^{a}} )} = {1 - \frac{{r - r^{a}}}{r_{c}}}},{{{r - r^{a}}} \leq r_{c}}} \\        {B = {{{e(0)}{{\mathbb{e}}^{H}(0)}} = 11^{H}}} \\        {{e(v)} = \lbrack {{1{\mathbb{e}}^{{j2\pi}\quad\frac{u}{\lambda}T^{a}}},\ldots\quad,{\mathbb{e}}^{{j2\pi}\quad{\frac{u}{\lambda}{\lbrack{N - 1}\rbrack}}T^{a}}} \rbrack^{T}} \\        {1 = \lbrack {1,\ldots\quad,1} \rbrack^{T}}        \end{matrix} & (12)        \end{matrix}$        where, without loss of generality, a common time-variant phase        term {e^(jθe)} on the mean components is set to zero, and I        denotes the identity matrix. Moreover, A is the ERAM signal        amplitude, σ² is the ERAM noise variance, and (•)^(H) denotes        the conjugate transpose operation. Lastly, without loss or        generality, a common time-variant phase term {e^(jθe)} on the        mean components is set to zero.

The Central Limit Theorem justifies the claim that z^(P) isasymptotically normal with mean and variance found from the Appendix:$\begin{matrix}{{ z^{P} \sim{\mathcal{N}( {\mu_{P},\sigma_{P}^{2}} )}}{\mu_{P} = {L( {{A^{2}{G^{2}( {r - r^{a}} )}{J^{2}( {v - v^{a}} )}} + {N\quad\sigma^{2}}} )}}{\sigma_{P}^{2} = {L\quad N\quad{\sigma^{2}( {{2A^{2}{G^{2}( {r - r^{a}} )}{J^{2}( {v - v^{a}} )}} + {N\quad\sigma^{2}}} )}}}{{J( {v - v^{a}} )} = \{ \begin{matrix}N & {v = v^{a}} \\\frac{\sin( {\pi\frac{\lbrack {v - v^{a}} \rbrack}{\lambda}T^{a}N} )}{\sin( {\pi\frac{\lbrack {v - v^{a}} \rbrack}{\lambda}T^{a}} )} & {v \neq v^{a}}\end{matrix} }} & (13)\end{matrix}$where the fact that: $\begin{matrix}\begin{matrix}{{{{{\mathbb{e}}^{H}( v^{\prime} )}{{\mathbb{e}}( v^{''} )}}}^{2} = {{{\mathbb{e}}^{H}( v^{\prime} )}{{\mathbb{e}}( v^{''} )}{{\mathbb{e}}^{H}( v^{''} )}{{\mathbb{e}}( v^{\prime} )}}} \\{= \lbrack \frac{\sin( {\pi\frac{\lbrack {v^{''} - v^{\prime}} \rbrack}{\lambda}T^{a}N} )}{\sin( {\pi\frac{\lbrack {v^{''} - v^{\prime}} \rbrack}{\lambda}T^{a}} )} \rbrack^{2}}\end{matrix} & (14)\end{matrix}$has been used.B.2.b Early-Late $\begin{matrix}{{z^{EL} = {\sum\limits_{\ell = 0}^{L - 1}{y_{\ell}^{H}{By}_{\ell}}}}{y_{\ell}\overset{IID}{\sim}{\mathcal{N}( {\mu_{y},K_{y}} )}}{\mu_{y} = {A\lbrack {{G( {r - r^{a_{E}}} )}{{\mathbb{e}}^{T}( {v - v^{a}} )}{G( {r - r^{a_{L}}} )}{{\mathbb{e}}^{T}( {v - v^{a}} )}} \rbrack}^{T}}{{K_{y} = {\sigma^{2}\begin{bmatrix}I & {\rho\quad I} \\{\rho\quad I} & I\end{bmatrix}}},{\rho = {1 - \frac{{r^{a_{E}} - r^{a_{L}}}}{r_{c}}}}}{B = {\begin{bmatrix}{{{\mathbb{e}}(0)}{{\mathbb{e}}^{H}(0)}} & 0 \\0 & {{- {{\mathbb{e}}(0)}}{{\mathbb{e}}^{H}(0)}}\end{bmatrix} = \begin{bmatrix}11^{H} & 0 \\0 & {- 11^{H}}\end{bmatrix}}}} & (15)\end{matrix}$where 0 is a matrix of zeros. The Early-Late measurement is distributedas: $\begin{matrix}{\quad{{ z^{EL} \sim{\mathcal{N}( {\mu_{EL},\sigma_{EL}^{2}} )}}\quad{\mu_{EL} = {{LA}^{2}{{J^{2}( {v - v^{a}} )}\lbrack {{G^{2}( {r - r^{a_{E}}} )} + {G^{2}( {r - r^{a_{L}}} )}} \rbrack}}}{\sigma_{EL}^{2} = {L\begin{pmatrix}{{2A^{2}N\quad\sigma^{2}{{J^{2}( {v - v^{a}} )}\begin{bmatrix}{{G^{2}( {r - r^{a_{E}}} )} + {G^{2}( {r - r^{a_{L}}} )} -} \\{2\rho\quad{G( {r - r^{a_{E}}} )}{G( {r - r^{a_{L}}} )}}\end{bmatrix}}} +} \\{2N^{2}{\sigma^{4}( {1 - \rho^{2}} )}}\end{pmatrix}}}}} & (16)\end{matrix}$B.2.c. High-Low. $\begin{matrix}{{z^{HL} = {\sum\limits_{\ell = 0}^{L - 1}{v_{\ell}^{H}B_{v_{\ell}}}}}{v_{\ell}\overset{IID}{\sim}{\mathcal{N}( {\mu_{v},K_{v}} )}}{\mu_{v} = {{{AG}( {r - r^{a_{P}}} )}\lbrack {{{\mathbb{e}}^{T}( {v - v^{a}} )}\quad{{\mathbb{e}}^{T}( {v - v^{a}} )}} \rbrack}^{T}}{K_{v} = {\sigma^{2}\begin{bmatrix}I & I \\I & I\end{bmatrix}}}{B = \begin{bmatrix}{{{\mathbb{e}}( \delta_{v} )}{{\mathbb{e}}^{H}( \delta_{v} )}} & 0 \\0 & {{- {{\mathbb{e}}( {- \delta_{v}} )}}{{\mathbb{e}}^{H}( {- \delta_{v}} )}}\end{bmatrix}}} & \quad & (17)\end{matrix}$The High-Low measurement is distributed as: $\begin{matrix}{\quad{{z^{HL} \sim {\mathcal{N}( {\mu_{HL},\sigma_{HL}^{2}} )}}\quad{\mu_{HL} = {{LA}^{2}{{G^{2}( {r - r^{a_{P}}} )}\lbrack {{J^{2}( {v - v^{a} - \delta_{v}} )} - {J^{2}( {v - v^{a} + \delta_{v}} )}} \rbrack}}}{\sigma_{HL}^{2} = {L\begin{pmatrix}{{2A^{2}{G^{2}( {r - r^{a_{P}}} )}{\sigma^{2}\begin{bmatrix}{{{NJ}^{2}( {v - v^{a} - \delta_{v}} )} + {{NJ}^{2}( {v - v^{a} + \delta_{v}} )} -} \\{2{J( {v - v^{a} - \delta_{v}} )}{J( {v - v^{a} + \delta_{v}} )}{J( {2\delta_{v}} )}}\end{bmatrix}}} +} \\{2{\sigma^{4}\lbrack {N^{2} - {J^{2}( {2\delta_{v}} )}} \rbrack}}\end{pmatrix}}}}} & (18)\end{matrix}$Thus, the nonlinear function related the state vector to the (noiseless)measurement vector is given as: $\begin{matrix}{{h( x_{k} )} = \begin{bmatrix}{{LA}_{k}^{2}{{J^{2}( {v_{k} - v_{k}^{a}} )}\lbrack {{G^{2}( {r_{k} - r_{k}^{a_{E}}} )} - {G^{2}( {r_{k} - r_{k}^{a_{L}}} )}} \rbrack}} \\{{LA}_{k}^{2}{{G^{2}( {v_{k} - v_{k}^{a_{P}}} )}\begin{bmatrix}{{J^{2}( {v_{k} - v_{k}^{a} - \delta_{v}} )} -} \\{J^{2}( {v_{k} - v_{k}^{a} + \delta_{v}} )}\end{bmatrix}}} \\{L\lbrack {{A_{k}^{2}{G^{2}( {r_{k} - r_{k}^{a}} )}{J^{2}( {v_{k} - v_{k}^{a}} )}} + {N\quad\sigma^{2}}} \rbrack}\end{bmatrix}} & (19)\end{matrix}$To determine the gradient Hκ, consider the following derivates:$\begin{matrix}\begin{matrix}\begin{matrix}{{\frac{\mathbb{d}}{\mathbb{d}r}{G^{2}( {r - r^{a}} )}} = {2\frac{\mathbb{d}}{\mathbb{d}r}{G( {r - r^{a}} )}}} \\{= \{ \begin{matrix}{{{- \frac{2}{r_{c}}}{G( {r - r^{a}} )}},} & {r \geq r^{a}} \\{{\frac{2}{r_{c}}{G( {r - r^{a}} )}},} & {r < r^{a}}\end{matrix} }\end{matrix} \\\begin{matrix}{{\frac{\mathbb{d}}{\mathbb{d}r}{J^{2}( {v - v^{a}} )}} = {2\frac{\mathbb{d}}{\mathbb{d}v}{J( {v - v^{a}} )}}} \\{= {\frac{2{J( {v - v^{a}} )}}{\sin( {\pi\frac{\lbrack {v - v^{a}} \rbrack}{\lambda}T^{a}} )} \times}} \\{\begin{bmatrix}{{\frac{\mathbb{d}}{\mathbb{d}v}{\sin( {\pi\frac{\lbrack {v - v^{a}} \rbrack}{\lambda}T^{a}N} )}} -} \\{{J( {v - v^{a}} )}\frac{\mathbb{d}}{\mathbb{d}v}{\sin( {\pi\frac{\lbrack {v - v^{a}} \rbrack}{\lambda}T^{a}} )}}\end{bmatrix}} \\{{= {\frac{2\pi\quad T^{a}{J( {v - v^{a}} )}}{\lambda\quad{\sin( {\pi\frac{\lbrack {v - v^{a}} \rbrack}{\lambda}T^{a}} )}}\begin{bmatrix}{{N\quad{\cos( {\pi\frac{\lbrack {v - v^{a}} \rbrack}{\lambda}T^{a}N} )}} -} \\{{J( {v - v^{a}} )} \times} \\{\cos( {\pi\frac{\lbrack {v - v^{a}} \rbrack}{\lambda}T^{a}} )}\end{bmatrix}}},} \\{v \neq v^{a}}\end{matrix} \\{{\frac{\mathbb{d}}{\mathbb{d}a}A^{2}} = {{\frac{\mathbb{d}}{\mathbb{d}a}10^{\frac{0}{5}}} = {\frac{\ln(10)}{5}A^{2}}}}\end{matrix} & (20)\end{matrix}$where it is noted that for${\upsilon = \upsilon^{a}},{{\frac{\mathbb{d}}{\mathbb{d}\upsilon}{J^{2}( {\upsilon - \upsilon^{a}} )}} = 0.}$The individual elements of the gradient can be written as:$\begin{matrix}{{{H_{k}^{({1,1})} = {{LA}_{k}^{2}{{J^{2}( {v_{k} - v_{k}^{a}} )}\lbrack  {\frac{\mathbb{d}}{\mathbb{d}r}{G^{2}( {r - r_{k}^{a_{E}}} )}} \middle| {}_{r = r_{k}}{{- \frac{\mathbb{d}}{\mathbb{d}r}}{G^{2}( {r - r_{k}^{a_{L}}} )}} |_{r = r_{k}} \rbrack}}}{H_{k}^{({1,2})} =  {{LA}_{k}^{2}\frac{\mathbb{d}}{\mathbb{d}v}{J^{2}( {v - v_{k}^{a}} )}} \middle| {}_{v = v_{k}}\lbrack {{G^{2}( {r_{k} - r_{k}^{a_{E}}} )} - {G^{2}( {r_{k} - r_{k}^{a_{L}}} )}} \rbrack }{H_{k}^{({1,3})} =  {L\frac{\mathbb{d}}{\mathbb{d}a}A_{k}^{2}} \middle| {}_{a = a_{k}}{{J^{2}( {v_{k} - v_{k}^{a}} )}\lbrack {{G^{2}( {r_{k} - r_{k}^{a_{E}}} )} - {G^{2}( {r_{k} - r_{k}^{a_{L}}} )}} \rbrack} }{H_{k}^{({2,1})} =  {{LA}_{k}^{2}\frac{\mathbb{d}}{\mathbb{d}r}{G^{2}( {r - r_{k}^{a_{P}}} )}} \middle| {}_{r = r_{k}}\lbrack {{J^{2}( {v_{k} - v_{k}^{a} - \delta_{v}} )} - {J^{2}( {v_{k} - v_{k}^{a} + \delta_{v}} )}} \rbrack }H_{k}^{({2,2})} = {{LA}_{k}^{2}{{G^{2}( {r_{k} - r_{k}^{a_{P}}} )}\lbrack {\frac{\mathbb{d}}{\mathbb{d}v}{J^{2}( {v - v_{k}^{a} - \delta_{v}} )}{_{v = v_{k}}{{- \frac{\mathbb{d}}{\mathbb{d}v}}{J^{2}( {v - v_{k}^{a} + \delta_{v}} )}}}_{v = v_{k}}} \rbrack}}}{H_{k}^{({2,3})} =  {L\frac{\mathbb{d}}{\mathbb{d}a}A_{k}^{2}} \middle| {}_{a = a_{k}}\quad{{G^{2}( {r_{k} - r_{k}^{a_{P}}} )}\lbrack \quad{{J^{2}( {v_{k} - v_{k}^{a} - \delta_{v}} )} - {J^{2}( {v_{k} - v_{k}^{a} + \delta_{v}} )}} \rbrack} }\quad{H_{k}^{({3,1})} = { {{LA}_{k}^{2}\frac{\mathbb{d}}{\mathbb{d}r}{G^{2}( {r - r_{k}^{a}} )}} \middle| {}_{r = r_{k}}{{J^{2}( {v_{k} - v_{k}^{a}} )}\quad H_{k}^{({3,2})}}  = { {{LA}_{k}^{2}{G^{2}( {r_{k} - r_{k}^{a}} )}\frac{\mathbb{d}}{\mathbb{d}v}{J^{2}( {v - v_{k}^{a}} )}} \middle| {}_{v = v_{k}}\quad H_{k}^{({3,3})}  =  {L\frac{\mathbb{d}}{\mathbb{d}a}A_{k}^{2}} \middle| {}_{a = a_{k}}{{G^{2}( {r_{k} - r_{k}^{a}} )}{J^{2}( {v_{k} - v_{k}^{a}} )}} }}}} & (21)\end{matrix}$where H_(k) ^((i−j)) is the element of H_(k) corresponding to the jthcolumn.

II. Filter

The extended Kalman filter (EKF) used to estimate the state vector foreach block is summarized below:

Initialization:{circumflex over (x)}_(−1|−1): derived from measurements obtained overseveral blocks.  (22)P_(−1|−1): derived from covariance based on â_(−1|−1).Prediction:{circumflex over (x)}_(k|k−1)=Φ{circumflex over (x)}_(k−1|k−1)  (23)Prediction Covariance:P _(k|k−1) =ΦP _(k−1|k−1) Φ′+Q  (24)Kalman Gain:K _(k) =P _(k|k−1) H _(k)′(H _(k) P _(k|k−1) H _(k) ′+R _(k))⁻¹  (25)Correction/Estimation:{circumflex over (x)} _(k|k) ={circumflex over (x)} _(k|k−1) +K _(k) [z_(k) −h({circumflex over (x)} _(k|k−1))]  (26)Estimation Covariance:{circumflex over (P)} _(k|k)=(I−K _(k) H _(k))P _(k|k−1)  (27)In support of the foregoing description, the following is a descriptionof the first and second order statistics of a Hermitian Quadratic Form.Consider a complex normal random column vector y of mean μ_(y) andcovariance Σ_(y):y˜CN(μ_(y),Σ_(y))  (28)Consider the indefinite quadratic form:z=y^(H)By, B^(H)=B  (29)where (•)^(H) denotes the conjugate tranpose, and B is a Hermitian,generally indefinite matrix.The mean and variance of z are sought:μ_(z)=E[z], σ_(z) ² E[(z−μ _(z))² ]=E[z ²]−μ_(z) ²  (30)where E[•] denotes the expected value. The mean may be written simplyas:μ_(z) =E[y ^(H) By]=tr(B·E[yy ^(H)])=μ_(y) ^(H) Bμ _(y) +tr(BΣ_(y))  (31)where tr(•) denotes the trace operation. To compute the variance, firstwrite the random vector in terms of its means and a zero-mean, randomcomponent:y=μ _(y) +w.  (32)Now, consider the non-central second moment: $\begin{matrix}\begin{matrix}{{E\lbrack z^{2} \rbrack} = {E\lbrack ( {y^{H}B\quad y} )^{2} \rbrack}} \\{= {E\lbrack ( {\lbrack {\mu_{y}^{H} + w^{H}} \rbrack{B\lbrack {\mu_{y} + w} \rbrack}} )^{2\quad} \rbrack}} \\{= {E\lbrack ( {{\mu_{y}^{H}B\quad\mu_{y}} + {w^{H}B\quad\mu_{y}} + {\mu_{y}^{H}B\quad w} + {w^{H}B\quad w}} )^{2} \rbrack}} \\{= {( {\mu_{y}^{H}B\quad\mu_{y}} )^{2} + {2\mu_{y}^{H}B\quad\mu_{y}{{tr}( {B\quad\Sigma_{y}} )}} + {2\mu_{y}^{H}B\quad\Sigma_{y}B\quad\mu_{y}} +}} \\{E\lbrack ( {w^{H}B\quad w} )^{2} \rbrack} \\{= {( {\mu_{y}^{H}B\quad\mu_{y}} )^{2} + {2\mu_{y}^{H}B\quad\mu_{y}{{tr}( {B\quad\Sigma_{y}} )}} + {2\mu_{y}^{H}B\quad\Sigma_{y}B\quad\mu_{y}} +}} \\{{{tr}^{2}( {B\quad\Sigma_{y}} )} + {{{tr}( \lbrack {B\quad\Sigma_{y}} \rbrack^{2} )}.}}\end{matrix} & (33)\end{matrix}$using (30), $\begin{matrix}\begin{matrix}{\sigma_{z}^{2} = {{E\lbrack z^{2} \rbrack} - ( {\mu_{y}^{H}B\quad\mu_{y}} )^{2} - {2\mu_{y}^{H}B\quad\mu_{y}{{tr}( {B\quad\Sigma_{y}} )}} - {{tr}^{2}( {B\quad\Sigma_{y}} )}}} \\{= {{2\mu_{y}^{H}B\quad\Sigma_{y}B\quad\mu_{y}} + {{tr}( \lbrack {B\quad\Sigma_{y}} \rbrack^{2} )}}}\end{matrix} & (34)\end{matrix}$

In further support of the foregoing description, the following providesa description of the procedure used to invert a 3×3 matrix. Consider anarbitrary three-by-three, real matrix AεR^(3×3). This descriptionprovides an explicit formula for the inverse of such a matrix. Theinverse of any N×N non-singular matrix may be expressed in terms of itsadjoint and determinant: $\begin{matrix}\begin{matrix}{A^{- 1} = \frac{{adj}(A)}{\det(A)}} \\{{{adj}(A)} = {{B\text{:}\quad b_{ij}} = {( {- 1} )^{i + j}{\det( A_{ij} )}}}} \\{{{\det(A)} = {\sum\limits_{i = 1}^{N}\quad{( {- 1} )^{i + j}a_{ij}{\det( A_{ij} )}}}},{\forall{j \in \{ {1,2,\ldots\quad,N} \}}}}\end{matrix} & (35)\end{matrix}$where a_(if) is the element of A corresponding to the ith row and thejth column, and A_(ij) is the sub-matrix with the ith row and the jthcolumn of A deleted.It turns out that for a 2×2 matrix, $\begin{matrix}{A_{2} = \begin{bmatrix}a_{11} & a_{12} \\a_{21} & a_{22}\end{bmatrix}} & (36)\end{matrix}$its determinant can be written explicitly as:det(A ₂)=a ₁₁ a ₂₂ −a ₁₂ a ₂₁  (37)while for a 3×3 matrix, $\begin{matrix}{A_{3} = \begin{bmatrix}a_{11} & a_{12} & a_{13} \\a_{21} & a_{22} & a_{23} \\a_{31} & a_{32} & a_{33}\end{bmatrix}} & (38)\end{matrix}$its determinant can be written explicitly as:det(A ₃₃)=a ₁₁ a ₂₂ a ₃₃ +a ₁₂ a ₂₃ a ₃₁ +a ₁₃ a ₂₁ a ₃₂ −a ₁₁ a ₂₃ a ₃₂−a ₁₂ a ₂₁ a ₃₃ −a ₁₂ a ₂₂ a ₃₁  (39)In case of interest, the matrix A₃ is symmetric:a_(ij)a_(ji), ∀iε{1,2, . . . , N}, ∀jε{1,2, . . . , N}  (40)In this case, it can be shown that: $\begin{matrix}\begin{matrix}{A_{3}^{- 3} = \frac{1}{\det( A_{3} )}} \\{\begin{bmatrix}{{a_{22}a_{33}} - a_{23}^{2}} & {- ( {{a_{12}a_{33}} - {a_{23}a_{13}}} )} & {{a_{12}a_{23}} - {a_{13}a_{22}}} \\{- ( {{a_{12}a_{33}} - {a_{23}a_{13}}} )} & {{a_{11}a_{33}} - a_{13}^{2}} & {- ( {{a_{11}a_{23}} - {a_{12}a_{13}}} )} \\{{a_{12}a_{23}} - {a_{13}a_{22}}} & {- ( {{a_{11}a_{23}} - {a_{12}a_{13}}} )} & {{a_{11}a_{22}} - a_{12}^{2}}\end{bmatrix}} \\{{\det( A_{3} )} = {{a_{11}a_{22}a_{33}} + {2\quad a_{12}a_{23}a_{13}} - {a_{11}a_{23}^{2}} - {a_{12}^{2}a_{33}} - {a_{13}^{2}a_{22}}}}\end{matrix} & (41)\end{matrix}$

While the foregoing is directed to embodiments of the present invention,other and further embodiments of the invention may be devised withoutdeparting from the basic scope thereof, and the scope thereof isdetermined by the claims that follow.

1. A method of tracking a signal comprising: (a) correlating a digitalsignal with a code using a hypothesis at a plurality of frequencies andat least one delay to produce correlation results; (b) measuring anenergy value of the correlation results; and (c) adjusting at least oneof the frequency or delay in response to the measured energy value toform the hypothesis of frequency and delay.
 2. The method of claim 1further comprising: (d) repeating steps (a), (b), (c) and (d) to improvethe measured energy value.
 3. The method of claim 1 further comprising:tracking, over time the measured energy values of the correlationresults; and improving the magnitude of the energy values usingadjustments of at least one of the frequency or delay.
 4. The method ofclaim 1 wherein the measured energy value form an energy vectorcomprising energy from a plurality of delays and frequencies.
 5. Themethod of claim 1 wherein the adjusting step is performed using adynamic model.
 5. The method of claim 5 wherein the dynamic model is anextended Kalman filter.
 7. The method of claim 1 wherein the correlatingstep is performed using a plurality of correlators measuring acorrelation result having delay as a variable, a plurality ofcorrelators measuring a correlation result having frequency as avariable, and a punctual correlator.
 8. The method of claim 7 whereinthe correlators define a multi-dimensional correlation spacerepresenting an energy amount within the correlation result.
 9. Themethod of claim 7 wherein the plurality of correlators measuring acorrelation result having delay as a variable comprise an earlycorrelator having a delay that is earlier than a delay of the punctualcorrelator and a late correlator having a delay that is later than adelay of the punctual correlator, and wherein the plurality ofcorrelators measuring a correlation result having frequency as avariable comprise a high correlator having a frequency that is higherthan a frequency of the punctual correlator and a low correlator havinga frequency that is lower than a frequency of the punctual correlator.10. The method of claim 1 wherein the energy value is a measured energyvector comprising measured quantities of a frequency and a delay. 11.The method of claim 10 wherein the measured quantities are used toupdate states of an extended Kalman filter.
 12. A method of tracking asignal comprising: (a) correlating a digital signal with a code using ahypothesis at a plurality of frequencies and at least one delay toproduce correlation results; (b) producing a measured energy vectorusing the correlation results, where the measured energy vectorcomprises a frequency quantity and a delay quantity; and (c) updating afrequency, and delay states of an extended Kalman filter in response tothe measured energy vector to form the hypothesis.
 13. The method ofclaim 12 further comprising: (d) repeating steps (a), (b), (c) and (d)to improve the magnitude of the measured energy vector.
 14. The methodof claim 12 further comprising: tracking, over time, the measured energyvector of the correlation results; and improving the magnitude of theenergy vector using adjustments of at least one of the frequency ordelay.
 15. The method of claim 12 wherein the correlating step isperformed using a plurality of correlators measuring a correlationresult having delay as a variable, a plurality of correlators measuringa correlation result having frequency as a variable, and a punctualcorrelator.
 16. The method of claim 15 wherein the correlators define amulti-dimensional correlation space representing an energy amount withinthe correlation result.
 17. The method of claim 15 wherein the pluralityof correlators measuring a correlation result having delay as a variablecomprise an early correlator having a delay that is earlier than a delayof the punctual correlator and a late correlator having a delay that islater than a delay of the punctual correlator, and wherein the pluralityof correlators measuring a correlation result having frequency as avariable comprise a high correlator having a frequency that is higherthan a frequency of the punctual correlator and a low correlator havinga frequency that is lower than a frequency of the punctual correlator.18. Apparatus for tracking a signal comprising: (a) a plurality ofcorrelators for correlating a digital signal with a code using ahypothesis at a plurality of frequencies and at least one delay toproduce correlation results; (b) a discriminator for measuring an energyvalue of the correlation results; (c) a feedback filter for adjusting atleast one of the frequency or delay in response to the measured energyvalue to form the hypothesis.
 19. The apparatus of claim 18 wherein thefeedback filter tracks, over time, the measured energy value of thecorrelation results; and improves the magnitude of the energy valueusing adjustments of at least one of the frequency or delay.
 20. Theapparatus of claim 19 wherein the measured energy forms an energy vectorcomprising energy from a plurality of delays and frequencies.
 21. Theapparatus of claim 19 wherein the plurality of correlators comprises aplurality of correlators measuring a correlation result having delay asa variable, a plurality of correlators measuring a correlation resulthaving frequency as a variable, and a punctual correlator.
 22. Theapparatus of claim 21 wherein the plurality of correlators define amulti-dimensional correlation space representing an energy amount withinthe correlation results.
 23. The apparatus of claim 21 wherein theplurality of correlators measuring a correlation result having delay asa variable comprise an early correlator having a delay that is earlierthan a delay of the punctual correlator and a late correlator having adelay that is later than a delay of the punctual correlator, and whereinthe plurality of correlators measuring a correlation result havingfrequency as a variable comprise a high correlator having a frequencythat is higher than a frequency of the punctual correlator and a lowcorrelator having a frequency that is lower than a frequency of thepunctual correlator.
 24. The apparatus of claim 19 wherein the energyvalue is a measured energy vector comprising a frequency quantity and adelay quantity.
 25. The apparatus of claim 24 wherein the measuredquantities are used to update states of an extended Kalman filter.